Conceptually, what's the difference between the discount rate and the intertemporal elasticity of substitution?

Suppose I have a standard maximization problem with a CRRA utility function $$\max_{c_t} \sum_{t=0}^\infty \beta^t \frac{c^{1-\sigma}}{1-\sigma}$$ It seems to me that both $$\beta$$ and $$\sigma$$ control how much we care about future consumption relative to current consumption.

Generally, in economics treat both of these parameters that are constant across time and across agents. Why do we use both? What's the conceptual difference between the two?

We can derive express this as $$\frac{c_{t+1}}{c_t}=\beta^{1/\sigma}$$

Does having a model with both and estimating these as separate objects make sense?

Also, I know that in CRRA the IES and the risk aversion are bundled up in one parameter, but there are utility functions that don't bundle these like Epstein–Zin - rather, I care about the difference between $$\beta$$ and IES.

Discount rate: $$\beta$$ measures how people discount future consumption relative to present. It represents how many utils you loose from delayed consumption. For example, if utility of consumption of apple at $$t=0$$ is 50, and $$\beta=0.5$$ then the discount rate tells us that the utility of consuming same apple at time $$t=1$$ would be 25.
The intuitive way of thinking about discount rate is that it is an unit adjustment. Present utility is more valuable than future utility, hence if you just focus on infratemporal utility you have in essence two units. Continuing with my previous example, the apple would give you 50 utils of present utility if consumed today, and 50 units of future utility if consumed tomorrow. $$\beta^t$$ is in essence a conversion rate for the utility at period $$t$$ so you can make intertemporal comparisons. Otherwise, it would be apples to oranges comparison.
Elasticity of Substitution: Measures how the ratio of present consumption, or continuing with the example of apples, how the ratio of present consumption of apples to future consumption of apples changes. In essence it tells you how many apples is the agent willing to give up today to have more apples tomorrow or vice versa. Note IES does not tell you by itself about the conversion rate between present and future utils, it directly answers question about consumption $$c_t$$.
These two measures are clearly not identical, save for special cases where $$\beta = IES$$ like with linear utility. Moreover, both of these measures answer different questions. Discount rate tells you what is the conversion rate between present and future utility, in fact this is also reflected by the fact that we actually usually put it in front of utility function i.e. general intertemporal utility formula is $$U = \sum \beta^t u(c_t)$$, whereas the elasticity answers the question of how many apples (not utils) is person willing to give up today to get more apples tomorrow.